EFFICIENT IIR NOTCH FILTER DESIGN VIA MULTIRATE FILTERING TARGETED AT HARMONIC DISTURBANCE REJECTION

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1 EFFICIENT IIR NOTCH FILTER DESIGN VIA MULTIRATE FILTERING TARGETED AT HARMONIC DISTURBANCE REJECTION Control Systems Tehnology group Tehnishe Universiteit Eindhoven Eindhoven, The Netherlands Dennis Bruijnen, René van de Molengraft and Maarten Steinbuh Abstrat: In this paper a omputationally heap method is proposed to filter a harmoni series where the first harmoni is a sine at the fundamental frequeny f and the super harmonis are sines at frequenies f, 3f, 4f... The method is based on multirate filter banks. With this method a large amount of IIR noth filters an be implemented muh more effiiently than by simply staggering single rate IIR noth filters without deteriorating the frequeny range of interest. Keywords: Noth filters, filter banks, harmoni disturbane rejetion. INTRODUCTION Periodi disturbanes are often found in pratial appliations. The field of repetitive ontrol (Hara et al., 988) deals with ontrol systems subjeted to periodi disturbanes and exploits the internal model priniple (Franis and Wonham, 975). Also in other researh areas, like eletri power systems, elimination of harmonis is addressed, (Qidwai and Bettayeb, 997; Rehka et al., 3; Czarneki and III, 5). Our interest lies in attenuating harmoni disturbane signals whih is a sublass of periodi disturbanes. The first harmoni is a sine at frequeny f, the so-alled fundamental frequeny, and the super harmonis are sines at the frequenies kf with k {, 3, 4,...}. Furthermore, it is assumed that the harmoni disturbanes ontain higher frequenies than the frequeny range of interest. Consider for example ontrol systems where the bandwidth frequeny lies in the lower frequeny range with respet to the sampling frequeny. Above the bandwidth frequeny, the only objetive is to attenuate disturbanes, whih does not restrit the sampling rate neessarily. The only requirement is that signals above the bandwidth frequeny do not alias suh that the signal ontent below the bandwidth is deteriorated. To attenuate harmonis at higher frequenies than the frequeny range of interest, a lowpass filter would be a natural hoie to remove these harmonis. However, the seletivity ouh filter is bad. The seletivity is defined as the slope of the magnitude response at the band edge. Signal ontent in the frequeny range of interest will either be deteriorated signifiantly or the periodi disturbane will not be suffiiently attenuated unless the differene between the frequeny range of interest and the disturbane frequeny is more than -3 deades. The use of noth filters is muh more desirable in that ase beause of their higher seletivity. Compared to the low pass filter, the filter order will inrease dramatially when nothes are plaed at eah harmoni frequeny resulting in high omputational osts. An alternative whih is often used in repetitive ontrol is a FIR omb filter, however, ompared to IIR filters, the required number of taps or order of the FIR filter is very

2 large to obtain equal seletivity properties, whih leads to high omputational osts. In this paper, an effiient method is proposed to implement a large number of IIR noth filters to remove harmonis. Effiient in the sense of good harmoni disturbane rejetion and in the sense of low omputational osts. The method is based on multirate filter banks (Strang and Nguyen, 996). The onept of multirate filtering has first been introdued in (Kan and Aggarwal, 97). A low order IIR filter an provide low deterioration at the frequeny range of interest together with good robustness against small frequeny variations of the harmoni frequenies. First, the struture and design of the filter bank will be presented. After that, some examples will be disussed. Finally, onlusions will be drawn.. METHOD. Multirate digital harmoni noth filter bank A digital filter with sampling rate onsisting of n nothes plaed at the frequenies f k = k n +, k =,,..., n () This reursive proess an be repeated resulting in an m-rate filter bank whih filters N = (n + ) m (6) harmonis (see also Table ) with frequenies f k = k (n + ) m, k =,,..., N (7) N m= m= m=3 m=4 n= n= n= n= Table. Total amount of harmoni frequenies N filtered by the m-rate filter bank ontaining n nothes at eah rate. To larify this reursive multirate temp proess, an example 5// is :4 shown in Fig.. In this example, attenuation of the first 5 harmonis is obtained by applying two 3rd order filters at a sampling rate of and 4 respetively. It is obvious that this signifiantly redues the omputational osts ompared to a 5th order filter at a sampling rate of. H m= Im suppresses n harmonis with the first harmoni at n+. The simplest onstrution ouh a digital harmoni noth (DHN) filter onsists of a zero at e jπ f k fs and a pole at ae jπ f k fs for k =,,..., n and a < for harmoni attenuation. If the same filter oeffiients are used at lower rates, the noth width beomes larger. To obtain an equal noth width for the ith-rate filter of an m-rate filter bank, it an be easily derived that the zeros and poles z k, p k, k =,,..., n should be H m=,m Nyquist frequeny (n+),m Im Re Re z k = e jπ f k,i () p k = a (n+)(i ) z k (3) with sampling rate of the ith-rate filter H total Nyquist frequeny Im,i = (n + ) i (4) If a signal is filtered with a DHN filter at a sampling rate of (i = ) and the output of this filter is filtered by a similar filter at n+ a sampling rate of (i = ), obtained by downsampling/deimation (Strang and Nguyen, 996), then a -rate DHN filter bank is obtained whih attenuates the harmoni frequenies f k = k (n + ), k =,,..., (n + ) (5),m (n+),m Fig.. Example of a multirate harmoni noth filter with n = 3 and m =. On the left the magnitude of H m=, H m= and H total = H m= H m= is shown. The equivalent single rate filter with sampling rate is shown on the right in terms of zeros o and poles x in the omplex plane. By hoosing values for, n and m, we obtain Re

3 the number of harmoni frequenies to be attenuated N the frequeny of the first harmoni f the output sampling frequeny,m whih are omputed by (4), (6) and (7) respetively. Suppose n =, then one noth is loated at the Nyquist frequeny (half the sampling rate). Aliasing will our if a disturbane frequeny is higher than the Nyquist frequeny. Fortunately, this noth filter keeps on fulfilling its task. This an be easily verified by onsidering a disturbane signal with a frequeny just above the Nyquist frequeny. Also for n >, aliasing harmonis whih oinide with noth frequenies will be filtered out.. Robustness to frequeny variations The distortion of the signal ontent of interest and the robustness to small frequeny variations of the harmonis is determined by the seletivity of the basi noth filter whih is multiplied for eah noth loation. The seletivity of the noth an be inreased by inreasing the order of the filter and/or using an Ellipti filter design approah (Thede, 996). A prototype noth filter an be reated by designing a low pass filter with the utoff frequeny lose to the Nyquist frequeny with the funtion ELLIP in MATLAB. The distane of the ut-off frequeny to the Nyquist frequeny determines the noth width and the distortion in the pass band. This prototype noth filter an then be shifted to all n frequenies (). For eah rate of the filter bank this proedure has to be arried out to obtain a multirate DHN filter as desribed before..3 Filter effiieny The omputational osts of the filter bank is assumed to be proportional to the amount of evaluations of the filter multiplied by the filter order, hene n + n n n (n + ) m = n m i= (n + ) i (8) Values for some n and m are shown in Table. With this omputational osts speifiation, the omputational osts N of a onventional sequene of nothes at a sampling rate beomes N N (9) m= m= m=3 m=4 n= n= n= n= Table. Costs expressed as the equivalent order at sampling rate. The filter effiieny improvement N (6), (8) and (9) N = (n + )m n m i= (n+) i From (4) and () follows that follows from = (n + ) m () N =,m () whih is a lear indiation of the filter effiieny improvement, i.e. it is proportional to the output sampling rate redution. In Table 3, the filter effiieny improvement is shown for some values of n and m. N m= m= m=3 m=4 n= 4 8 n= n= n= Table 3. Filter effiieny improvement.4 Design proedure Now, the requirements that limit the hoie for, n and m are disussed. First of all, eah hardware platform has its maximum sampling rate. Furthermore, the fundamental frequeny f of the harmoni disturbane and the number of harmonis are given. Next, signal ontent of interest at low frequenies f low should be outputted with suffiient timing resolution, e.g.,m > f low. All ombinations of n, m and fulfilling these onstraints an be heked. The parameter ombination with the largest N value is the most effiient implementation..4.. Example Suppose that the highest frequeny of interest f low = Hz, the fundamental frequeny of the harmonis is f = Hz and the number of harmonis to be suppressed is 4. Thus, the harmoni frequenies are f = Hz, f = 4 Hz, f 3 = 6 Hz,..., f 4 = 48 Hz. Furthermore, assume that the maximum sampling rate is 6 Hz and the minimum output sampling rate is,m f low.

4 tempimage temp 5//3 4 Then, the optimum values for n, m and are, 3, 54 Hz respetively, implying an output sampling rate of,m = 6 Hz, the number of noth filters N = 6 and an effiieny improvement N of Example In this example is shown that the algorithm is also effetive for a totally different hoie of parameters. Furthermore, it is shown that the filter still works if the effiieny inrease is very large. Finally, it is shown that a noth loated at the Nyquist frequeny filters disturbanes just as good as a noth below the Nyquistfrequeny like in example. For f low = Hz, f = 3 Hz, 5 harmonis, 5 Hz and,m f low the optimum values for n, m and are, 7, 384 Hz respetively. These values imply N = 7, N = 64 and,m = Hz. In this ase, the multirate DHN filter is 64 times more effiient than the equivalent singlerate filter. Suh large effiieny gains an be ahieved if f low and f are small with respet to the sampling rate. 3. Example The filter input signal x = sin πf low t + with stohasti variable 3. SIMULATIONS N sin(πkf t + S k ) () k= S k U(, π) is subjeted to the filter as desribed in Setion.4. with n =, m = 3 and = 54 Hz. For the basi filter, a nd order Ellipti filter with a pass band ripple of db and an attenuation of 8 db is used. The frequeny response magnitude for all rates are shown in Fig.. Also the equivalent single rate filter is shown in Fig.. The result of filtering one realization of () with the multirate filter is shown in Fig. 3 and the result of filtering the same realization of () with the equivalent single rate filter is shown in Fig. 4. Comparing Fig. 3 and Fig. 4, we see that the is nearly the same, only the output sampling rate is lower for the multirate filter. In Fig. 5, the step response of the multirate filter and the equivalent singlerate filter is ompared. It an be observed that the multirate filter has less overshoot than the singlerate filter. All other simulations onduted, showed the same result with varying filter settings. single rate rate 3 rate rate tempimage temp 5//3 4: frequeny [Hz] Fig.. Frequeny response magnitude of all filters in the multirate filter, rate : 54 Hz, rate : 8 Hz, rate 3: 6 Hz. The equivalent single rate filter is shown in the bottom figure tempimage temp 5//3 4: Fig. 3. Result of filtering a realization of () using the multirate filter Fig. 4. Result of filtering a realization of () using the equivalent single rate filter. If f is inreased by. % in (), then the result as shown in Fig. 6 is obtained. The performane has dereased somewhat, however, there is still

5 tempimage temp 5//3 tempimage temp 5//3 4:46 page # multirate filter singlerate filter tempimage temp 5//3 4:43 page # Fig. 5. Step response of the multirate and equivalent singlerate filter. a lear separation of the sine at f low and the harmoni disturbanes single rate rate 7 rate 6 rate 5 rate 4 rate 3 rate rate Fig. 6. Result of filtering a realization of () with a. % frequeny shift of f using the multirate filter. 3. Example Next, the example of Setion.4. is onsidered where n, m and are, 7 and 384 Hz respetively. For the basi filter, a nd order Ellipti filter is used with a pass band ripple of 3 db and a stop band ripple of 8 db. The frequeny response magnitude of the 3 rates of this filter and the equivalent single rate filter are shown in Fig. 7. Again, () is subjeted to the filter with f low = Hz and f = 3 Hz, so the input signal is a sine at Hz with 7 harmonis with the same amplitude. The results are shown in Fig. 8. frequeny [Hz] Fig. 7. Frequeny response magnitude of all filters in the multirate filter, rate : 384 Hz, rate : 9 Hz, rate 3: 96 Hz, rate 4: 48 Hz, rate 5: 4 Hz, rate 6: Hz, rate 7: 6 Hz. The equivalent single rate filter is shown in the bottom figure. 4. CONCLUSIONS It has been shown that the multirate DHN filter performs just as good as the equivalent single rate filter, however with muh lower omputational osts at the ost of a redued output sampling rate. It was found that the omputational osts redution is proportional to the ratio of the sampling rate and the output sampling rate. Furthermore, the multirate DHN filter shows less overshoot than the equivalent single rate filter.

6 tempimage temp 5//3 3: page # Fig. 8. Result of filtering a realization of () using the multirate DHN filter. 5. REFERENCES Czarneki, L.S. and H.L. Ginn III (5). The effet of the design method on effiieny of resonant harmoni filters. In: Power delivery. Vol.. pp Franis, B.A. and W.M. Wonham (975). The internal model priniple for linear multivariable regulators. Applied Mathematis and Optis, Hara, S., Y. Yamamoto, T. Omata and M. Nakano (988). Repetitive ontrol system - a new type servo system. IEEE Transations on Automati Control 33, Kan, E. and J. Aggarwal (97). Multirate digital filtering. Audio and Eletroaoustis, 3 5. Qidwai, S.U.A. and M. Bettayeb (997). A new robust sheme for harmoni elimination. Eletri Power Systems Researh 43, Rehka, S., E. Ngandui, J. Xu and P. Siard (3). Performane evaluation of harmonis detetion methods applied to harmonis ompensation in presene of ommon power quality problems. Mathematis and Computers in Simulation 63, Strang, G. and T. Nguyen (996). Wavelets and Filter Banks. Cambridge Press. Wellesley. Thede, L. (996). Analog and digital filter design using C. Prentie-Hall, In.. Upper Saddle River, NJ, USA.